Optimal. Leaf size=598 \[ -\frac {i \left (a+b \tan ^{-1}(c x)\right )^2}{3 c^3 e}-\frac {2 b \log \left (\frac {2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{3 c^3 e}-\frac {d \left (a+b \tan ^{-1}(c x)\right )^2}{2 c^2 e^2}-\frac {i b d^3 \text {Li}_2\left (1-\frac {2}{1-i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{e^4}+\frac {i b d^3 \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{e^4}+\frac {d^3 \log \left (\frac {2}{1-i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )^2}{e^4}-\frac {d^3 \left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac {2 c (d+e x)}{(1-i c x) (c d+i e)}\right )}{e^4}+\frac {d^2 x \left (a+b \tan ^{-1}(c x)\right )^2}{e^3}+\frac {i d^2 \left (a+b \tan ^{-1}(c x)\right )^2}{c e^3}+\frac {2 b d^2 \log \left (\frac {2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{c e^3}-\frac {d x^2 \left (a+b \tan ^{-1}(c x)\right )^2}{2 e^2}+\frac {a b d x}{c e^2}+\frac {x^3 \left (a+b \tan ^{-1}(c x)\right )^2}{3 e}-\frac {b x^2 \left (a+b \tan ^{-1}(c x)\right )}{3 c e}-\frac {i b^2 \text {Li}_2\left (1-\frac {2}{i c x+1}\right )}{3 c^3 e}-\frac {b^2 \tan ^{-1}(c x)}{3 c^3 e}-\frac {b^2 d \log \left (c^2 x^2+1\right )}{2 c^2 e^2}+\frac {b^2 x}{3 c^2 e}+\frac {b^2 d^3 \text {Li}_3\left (1-\frac {2}{1-i c x}\right )}{2 e^4}-\frac {b^2 d^3 \text {Li}_3\left (1-\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{2 e^4}+\frac {i b^2 d^2 \text {Li}_2\left (1-\frac {2}{i c x+1}\right )}{c e^3}+\frac {b^2 d x \tan ^{-1}(c x)}{c e^2} \]
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Rubi [A] time = 0.67, antiderivative size = 598, normalized size of antiderivative = 1.00, number of steps used = 23, number of rules used = 13, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.619, Rules used = {4876, 4846, 4920, 4854, 2402, 2315, 4852, 4916, 260, 4884, 321, 203, 4858} \[ -\frac {i b d^3 \text {PolyLog}\left (2,1-\frac {2}{1-i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{e^4}+\frac {i b d^3 \left (a+b \tan ^{-1}(c x)\right ) \text {PolyLog}\left (2,1-\frac {2 c (d+e x)}{(1-i c x) (c d+i e)}\right )}{e^4}-\frac {i b^2 \text {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{3 c^3 e}+\frac {b^2 d^3 \text {PolyLog}\left (3,1-\frac {2}{1-i c x}\right )}{2 e^4}-\frac {b^2 d^3 \text {PolyLog}\left (3,1-\frac {2 c (d+e x)}{(1-i c x) (c d+i e)}\right )}{2 e^4}+\frac {i b^2 d^2 \text {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{c e^3}-\frac {d \left (a+b \tan ^{-1}(c x)\right )^2}{2 c^2 e^2}-\frac {i \left (a+b \tan ^{-1}(c x)\right )^2}{3 c^3 e}-\frac {2 b \log \left (\frac {2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{3 c^3 e}+\frac {d^2 x \left (a+b \tan ^{-1}(c x)\right )^2}{e^3}+\frac {i d^2 \left (a+b \tan ^{-1}(c x)\right )^2}{c e^3}+\frac {d^3 \log \left (\frac {2}{1-i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )^2}{e^4}-\frac {d^3 \left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac {2 c (d+e x)}{(1-i c x) (c d+i e)}\right )}{e^4}+\frac {2 b d^2 \log \left (\frac {2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{c e^3}-\frac {d x^2 \left (a+b \tan ^{-1}(c x)\right )^2}{2 e^2}+\frac {a b d x}{c e^2}+\frac {x^3 \left (a+b \tan ^{-1}(c x)\right )^2}{3 e}-\frac {b x^2 \left (a+b \tan ^{-1}(c x)\right )}{3 c e}-\frac {b^2 d \log \left (c^2 x^2+1\right )}{2 c^2 e^2}+\frac {b^2 x}{3 c^2 e}-\frac {b^2 \tan ^{-1}(c x)}{3 c^3 e}+\frac {b^2 d x \tan ^{-1}(c x)}{c e^2} \]
Antiderivative was successfully verified.
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Rule 203
Rule 260
Rule 321
Rule 2315
Rule 2402
Rule 4846
Rule 4852
Rule 4854
Rule 4858
Rule 4876
Rule 4884
Rule 4916
Rule 4920
Rubi steps
\begin {align*} \int \frac {x^3 \left (a+b \tan ^{-1}(c x)\right )^2}{d+e x} \, dx &=\int \left (\frac {d^2 \left (a+b \tan ^{-1}(c x)\right )^2}{e^3}-\frac {d x \left (a+b \tan ^{-1}(c x)\right )^2}{e^2}+\frac {x^2 \left (a+b \tan ^{-1}(c x)\right )^2}{e}-\frac {d^3 \left (a+b \tan ^{-1}(c x)\right )^2}{e^3 (d+e x)}\right ) \, dx\\ &=\frac {d^2 \int \left (a+b \tan ^{-1}(c x)\right )^2 \, dx}{e^3}-\frac {d^3 \int \frac {\left (a+b \tan ^{-1}(c x)\right )^2}{d+e x} \, dx}{e^3}-\frac {d \int x \left (a+b \tan ^{-1}(c x)\right )^2 \, dx}{e^2}+\frac {\int x^2 \left (a+b \tan ^{-1}(c x)\right )^2 \, dx}{e}\\ &=\frac {d^2 x \left (a+b \tan ^{-1}(c x)\right )^2}{e^3}-\frac {d x^2 \left (a+b \tan ^{-1}(c x)\right )^2}{2 e^2}+\frac {x^3 \left (a+b \tan ^{-1}(c x)\right )^2}{3 e}+\frac {d^3 \left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac {2}{1-i c x}\right )}{e^4}-\frac {d^3 \left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{e^4}-\frac {i b d^3 \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1-i c x}\right )}{e^4}+\frac {i b d^3 \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{e^4}+\frac {b^2 d^3 \text {Li}_3\left (1-\frac {2}{1-i c x}\right )}{2 e^4}-\frac {b^2 d^3 \text {Li}_3\left (1-\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{2 e^4}-\frac {\left (2 b c d^2\right ) \int \frac {x \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx}{e^3}+\frac {(b c d) \int \frac {x^2 \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx}{e^2}-\frac {(2 b c) \int \frac {x^3 \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx}{3 e}\\ &=\frac {i d^2 \left (a+b \tan ^{-1}(c x)\right )^2}{c e^3}+\frac {d^2 x \left (a+b \tan ^{-1}(c x)\right )^2}{e^3}-\frac {d x^2 \left (a+b \tan ^{-1}(c x)\right )^2}{2 e^2}+\frac {x^3 \left (a+b \tan ^{-1}(c x)\right )^2}{3 e}+\frac {d^3 \left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac {2}{1-i c x}\right )}{e^4}-\frac {d^3 \left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{e^4}-\frac {i b d^3 \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1-i c x}\right )}{e^4}+\frac {i b d^3 \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{e^4}+\frac {b^2 d^3 \text {Li}_3\left (1-\frac {2}{1-i c x}\right )}{2 e^4}-\frac {b^2 d^3 \text {Li}_3\left (1-\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{2 e^4}+\frac {\left (2 b d^2\right ) \int \frac {a+b \tan ^{-1}(c x)}{i-c x} \, dx}{e^3}+\frac {(b d) \int \left (a+b \tan ^{-1}(c x)\right ) \, dx}{c e^2}-\frac {(b d) \int \frac {a+b \tan ^{-1}(c x)}{1+c^2 x^2} \, dx}{c e^2}-\frac {(2 b) \int x \left (a+b \tan ^{-1}(c x)\right ) \, dx}{3 c e}+\frac {(2 b) \int \frac {x \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx}{3 c e}\\ &=\frac {a b d x}{c e^2}-\frac {b x^2 \left (a+b \tan ^{-1}(c x)\right )}{3 c e}+\frac {i d^2 \left (a+b \tan ^{-1}(c x)\right )^2}{c e^3}-\frac {d \left (a+b \tan ^{-1}(c x)\right )^2}{2 c^2 e^2}-\frac {i \left (a+b \tan ^{-1}(c x)\right )^2}{3 c^3 e}+\frac {d^2 x \left (a+b \tan ^{-1}(c x)\right )^2}{e^3}-\frac {d x^2 \left (a+b \tan ^{-1}(c x)\right )^2}{2 e^2}+\frac {x^3 \left (a+b \tan ^{-1}(c x)\right )^2}{3 e}+\frac {d^3 \left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac {2}{1-i c x}\right )}{e^4}+\frac {2 b d^2 \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1+i c x}\right )}{c e^3}-\frac {d^3 \left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{e^4}-\frac {i b d^3 \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1-i c x}\right )}{e^4}+\frac {i b d^3 \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{e^4}+\frac {b^2 d^3 \text {Li}_3\left (1-\frac {2}{1-i c x}\right )}{2 e^4}-\frac {b^2 d^3 \text {Li}_3\left (1-\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{2 e^4}-\frac {\left (2 b^2 d^2\right ) \int \frac {\log \left (\frac {2}{1+i c x}\right )}{1+c^2 x^2} \, dx}{e^3}+\frac {\left (b^2 d\right ) \int \tan ^{-1}(c x) \, dx}{c e^2}+\frac {b^2 \int \frac {x^2}{1+c^2 x^2} \, dx}{3 e}-\frac {(2 b) \int \frac {a+b \tan ^{-1}(c x)}{i-c x} \, dx}{3 c^2 e}\\ &=\frac {a b d x}{c e^2}+\frac {b^2 x}{3 c^2 e}+\frac {b^2 d x \tan ^{-1}(c x)}{c e^2}-\frac {b x^2 \left (a+b \tan ^{-1}(c x)\right )}{3 c e}+\frac {i d^2 \left (a+b \tan ^{-1}(c x)\right )^2}{c e^3}-\frac {d \left (a+b \tan ^{-1}(c x)\right )^2}{2 c^2 e^2}-\frac {i \left (a+b \tan ^{-1}(c x)\right )^2}{3 c^3 e}+\frac {d^2 x \left (a+b \tan ^{-1}(c x)\right )^2}{e^3}-\frac {d x^2 \left (a+b \tan ^{-1}(c x)\right )^2}{2 e^2}+\frac {x^3 \left (a+b \tan ^{-1}(c x)\right )^2}{3 e}+\frac {d^3 \left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac {2}{1-i c x}\right )}{e^4}+\frac {2 b d^2 \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1+i c x}\right )}{c e^3}-\frac {2 b \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1+i c x}\right )}{3 c^3 e}-\frac {d^3 \left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{e^4}-\frac {i b d^3 \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1-i c x}\right )}{e^4}+\frac {i b d^3 \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{e^4}+\frac {b^2 d^3 \text {Li}_3\left (1-\frac {2}{1-i c x}\right )}{2 e^4}-\frac {b^2 d^3 \text {Li}_3\left (1-\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{2 e^4}+\frac {\left (2 i b^2 d^2\right ) \operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+i c x}\right )}{c e^3}-\frac {\left (b^2 d\right ) \int \frac {x}{1+c^2 x^2} \, dx}{e^2}-\frac {b^2 \int \frac {1}{1+c^2 x^2} \, dx}{3 c^2 e}+\frac {\left (2 b^2\right ) \int \frac {\log \left (\frac {2}{1+i c x}\right )}{1+c^2 x^2} \, dx}{3 c^2 e}\\ &=\frac {a b d x}{c e^2}+\frac {b^2 x}{3 c^2 e}-\frac {b^2 \tan ^{-1}(c x)}{3 c^3 e}+\frac {b^2 d x \tan ^{-1}(c x)}{c e^2}-\frac {b x^2 \left (a+b \tan ^{-1}(c x)\right )}{3 c e}+\frac {i d^2 \left (a+b \tan ^{-1}(c x)\right )^2}{c e^3}-\frac {d \left (a+b \tan ^{-1}(c x)\right )^2}{2 c^2 e^2}-\frac {i \left (a+b \tan ^{-1}(c x)\right )^2}{3 c^3 e}+\frac {d^2 x \left (a+b \tan ^{-1}(c x)\right )^2}{e^3}-\frac {d x^2 \left (a+b \tan ^{-1}(c x)\right )^2}{2 e^2}+\frac {x^3 \left (a+b \tan ^{-1}(c x)\right )^2}{3 e}+\frac {d^3 \left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac {2}{1-i c x}\right )}{e^4}+\frac {2 b d^2 \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1+i c x}\right )}{c e^3}-\frac {2 b \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1+i c x}\right )}{3 c^3 e}-\frac {d^3 \left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{e^4}-\frac {b^2 d \log \left (1+c^2 x^2\right )}{2 c^2 e^2}-\frac {i b d^3 \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1-i c x}\right )}{e^4}+\frac {i b^2 d^2 \text {Li}_2\left (1-\frac {2}{1+i c x}\right )}{c e^3}+\frac {i b d^3 \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{e^4}+\frac {b^2 d^3 \text {Li}_3\left (1-\frac {2}{1-i c x}\right )}{2 e^4}-\frac {b^2 d^3 \text {Li}_3\left (1-\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{2 e^4}-\frac {\left (2 i b^2\right ) \operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+i c x}\right )}{3 c^3 e}\\ &=\frac {a b d x}{c e^2}+\frac {b^2 x}{3 c^2 e}-\frac {b^2 \tan ^{-1}(c x)}{3 c^3 e}+\frac {b^2 d x \tan ^{-1}(c x)}{c e^2}-\frac {b x^2 \left (a+b \tan ^{-1}(c x)\right )}{3 c e}+\frac {i d^2 \left (a+b \tan ^{-1}(c x)\right )^2}{c e^3}-\frac {d \left (a+b \tan ^{-1}(c x)\right )^2}{2 c^2 e^2}-\frac {i \left (a+b \tan ^{-1}(c x)\right )^2}{3 c^3 e}+\frac {d^2 x \left (a+b \tan ^{-1}(c x)\right )^2}{e^3}-\frac {d x^2 \left (a+b \tan ^{-1}(c x)\right )^2}{2 e^2}+\frac {x^3 \left (a+b \tan ^{-1}(c x)\right )^2}{3 e}+\frac {d^3 \left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac {2}{1-i c x}\right )}{e^4}+\frac {2 b d^2 \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1+i c x}\right )}{c e^3}-\frac {2 b \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1+i c x}\right )}{3 c^3 e}-\frac {d^3 \left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{e^4}-\frac {b^2 d \log \left (1+c^2 x^2\right )}{2 c^2 e^2}-\frac {i b d^3 \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1-i c x}\right )}{e^4}+\frac {i b^2 d^2 \text {Li}_2\left (1-\frac {2}{1+i c x}\right )}{c e^3}-\frac {i b^2 \text {Li}_2\left (1-\frac {2}{1+i c x}\right )}{3 c^3 e}+\frac {i b d^3 \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{e^4}+\frac {b^2 d^3 \text {Li}_3\left (1-\frac {2}{1-i c x}\right )}{2 e^4}-\frac {b^2 d^3 \text {Li}_3\left (1-\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{2 e^4}\\ \end {align*}
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Mathematica [B] time = 22.33, size = 1413, normalized size = 2.36 \[ \text {result too large to display} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.58, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b^{2} x^{3} \arctan \left (c x\right )^{2} + 2 \, a b x^{3} \arctan \left (c x\right ) + a^{2} x^{3}}{e x + d}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 50.03, size = 2136, normalized size = 3.57 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {1}{6} \, a^{2} {\left (\frac {6 \, d^{3} \log \left (e x + d\right )}{e^{4}} - \frac {2 \, e^{2} x^{3} - 3 \, d e x^{2} + 6 \, d^{2} x}{e^{3}}\right )} + \frac {2 \, e^{3} \int \frac {36 \, {\left (b^{2} c^{2} e^{3} x^{5} + b^{2} e^{3} x^{3}\right )} \arctan \left (c x\right )^{2} + 3 \, {\left (b^{2} c^{2} e^{3} x^{5} + b^{2} e^{3} x^{3}\right )} \log \left (c^{2} x^{2} + 1\right )^{2} + 4 \, {\left (24 \, a b c^{2} e^{3} x^{5} - 2 \, b^{2} c e^{3} x^{4} - 3 \, b^{2} c d^{2} e x^{2} - 6 \, b^{2} c d^{3} x + {\left (b^{2} c d e^{2} + 24 \, a b e^{3}\right )} x^{3}\right )} \arctan \left (c x\right ) + 2 \, {\left (2 \, b^{2} c^{2} e^{3} x^{5} - b^{2} c^{2} d e^{2} x^{4} + 3 \, b^{2} c^{2} d^{2} e x^{3} + 6 \, b^{2} c^{2} d^{3} x^{2}\right )} \log \left (c^{2} x^{2} + 1\right )}{c^{2} e^{4} x^{3} + c^{2} d e^{3} x^{2} + e^{4} x + d e^{3}}\,{d x} + 4 \, {\left (2 \, b^{2} e^{2} x^{3} - 3 \, b^{2} d e x^{2} + 6 \, b^{2} d^{2} x\right )} \arctan \left (c x\right )^{2} - {\left (2 \, b^{2} e^{2} x^{3} - 3 \, b^{2} d e x^{2} + 6 \, b^{2} d^{2} x\right )} \log \left (c^{2} x^{2} + 1\right )^{2}}{96 \, e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {x^3\,{\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )}^2}{d+e\,x} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{3} \left (a + b \operatorname {atan}{\left (c x \right )}\right )^{2}}{d + e x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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